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Chapter 4: Problem 76

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is in the correct mode.) $$\sec 225^{\circ}$$

Short Answer

Expert verified
The value of \( \sec 225^{\circ} \) rounded off to four decimal places is -1.4142.

Step by step solution

01

Set calculator to degree mode

Before starting the calculation, ensure that the calculator is in degree mode as the angle is given in degrees. This setting is required as the value for trigonometric functions varies based on whether the angle is in degrees or radians.
02

Calculate cosine of the angle

Calculate the cosine of the angle in degrees, i.e., calculate \( \cos 225^{\circ} \).
03

Find the secant

The secant of an angle is the reciprocal of the cosine of the angle. So, find the reciprocal of \( \cos 225^{\circ} \) to find \( \sec 225^{\circ} \).
04

Round the answer

The last step is to round off the result to four decimal places as instructed in the question.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree Mode
In trigonometry, angle measurements can be expressed in degrees or radians. When using a calculator to solve problems involving angles measured in degrees, it's crucial to switch the calculator to degree mode. This ensures that the calculations reflect the correct angle units. Degree mode is essential when dealing with angles like 225掳 because the trigonometric values will significantly differ if the calculator is set to radians.
  • Most scientific calculators have a mode button that allows you to toggle between degrees and radians. Look for indications like 'DEG' to confirm the degree setting.
  • Never forget this step as it can drastically change the outcome of trigonometric function evaluations.
  • Always verify the mode before starting any trigonometric calculations involving degrees.
Secant Function
The secant function, denoted as \( \sec \theta \), is a reciprocal trigonometric function derived from the cosine function. For a given angle \( \theta \), the secant is defined as the reciprocal of the cosine:\[ \sec \theta = \frac{1}{\cos \theta} \]
  • Secant is less commonly used than sine or cosine and is often introduced after these more fundamental trigonometric functions.
  • Keep in mind, the secant function is not defined when \( \cos \theta = 0 \), because division by zero is undefined in mathematics.
  • It is important to know the cosine value first to determine the secant correctly.
For example, to find \( \sec 225^{\circ} \), you first need \( \cos 225^{\circ} \), which you then take the reciprocal of to obtain the secant.
Cosine Function
The cosine function is one of the primary trigonometric functions. It is abbreviated as \( \cos \theta \) and represents the ratio of the adjacent side to the hypotenuse in a right triangle. When dealing with angles like 225掳, cosine helps find out the related trigonometric values.
  • The function is periodic with a cycle of 360掳 or \( 2\pi \) radians. This means it repeats its pattern every 360掳 on the unit circle.
  • Knowing key points on the unit circle, like 0掳, 90掳, 180掳, 270掳, and 360掳, helps in understanding cosine's behavior.
  • At 225掳, \( \cos 225^{\circ} \) is typically negative because it lies in the third quadrant of the unit circle where both sine and cosine are negative.
Angle Measurement
Angle measurement is a critical concept in trigonometry that defines how angles are calculated and interpreted. There are two main units for measuring angles: degrees and radians. Degrees are more common in everyday use and in educational settings. When working with trigonometric functions, the angle measurement must be accurate and consistent with your calculator's settings.
  • A full circle is divided into 360 degrees or \( 2\pi \) radians.
  • Understand the conversion between degrees and radians: \( 180^\circ = \pi \) radians.
  • Angles are usually measured counterclockwise from the positive x-axis on the unit circle. This consistent starting point helps in understanding trigonometric values across different quadrants.
This becomes particularly important in problems like evaluating \( \sec 225^{\circ} \), where understanding the precise position on the unit circle gives insight into the signs and values of trigonometric functions.

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Most popular questions from this chapter

The numbers of hours \(H\) of daylight in Denver, Colorado, on the 15 th of each month are: \(1(9.67), 2(10.72), 3(11.92), 4(13.25)\) \(5(14.37), \quad 6(14.97), \quad 7(14.72), \quad 8(13.77), \quad 9(12.48)\) \(10(11.18), \quad 11(10.00), \quad 12(9.38) . \quad\) The month is represented by \(t,\) with \(t=1\) corresponding to January. A model for the data is $$H(t)=12.13+2.77 \sin \left(\frac{\pi t}{6}-1.60\right)$$. (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.

Determine whether the statement is true or false. Justify your answer. $$\sec 30^{\circ}=\csc 60^{\circ}$$

(a) Complete the table. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 0^{\circ} & 18^{\circ} & 36^{\circ} & 54^{\circ} & 72^{\circ} & 90^{\circ} \\ \hline \sin \theta & & & & & & \\ \hline \cos \theta & & & & & & \\ \hline \end{array}$$ (b) Discuss the behavior of the sine function for \(\theta\) in the range from \(0^{\circ}\) to \(90^{\circ} .\) (c) Discuss the behavior of the cosine function for \(\theta\) in the range from \(0^{\circ}\) to \(90^{\circ} .\) (d) Use the definitions of the sine and cosine functions to explain the results of parts (b) and (c).

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) \(\mathrm{As} x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) \(\mathrm{As} x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the interval \(-2 \pi, 2 \pi .\) Use the graphs to find real numbers \(x\) such that \(y_{1}=y_{2}\). $$\begin{aligned} &y_{1}=\sin x\\\ &y_{2}=-\frac{1}{2} \end{aligned}$$
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